Plastic mechanics, also known as plastic theory, is a branch of solid mechanics. This paper mainly studies the relationship between plastic deformation and external force, as well as the stress field, strain field and related laws inside the object, and its corresponding numerical analysis method. After an object is subjected to a large enough external force, part or all of its deformation will go beyond the elastic range and enter a plastic state. After the external force is removed, part or all of the deformation will not disappear, and the object cannot be completely restored to its original state. It should be noted that the permanent deformation considered in plastic mechanics is only related to the stress-strain history and does not change with time. The time-related part of permanent deformation belongs to the category of rheological research.
General plastic mechanics is divided into mathematical plastic mechanics and applied plastic mechanics, and its meaning is similar to mathematical elastic theory and applied elastic mechanics. The former is a classical precise theory, while the latter is a highly applied theory based on various assumptions of the former and the needs of practical application, plus some supplementary simplified assumptions. Mathematically, the application of plastic mechanics is rough, but from the application point of view, its equation and calculation formula are relatively simple, which can meet many structural design requirements.
Main contents of plastic mechanics
From the process of discipline establishment, plastic mechanics is based on experiments, from which the deformation law of stressed objects exceeding the elastic limit is found. On this basis, reasonable assumptions and simplified models are put forward, the constitutive relationship of materials after stress exceeds the elastic limit is determined, and the basic equations of plastic mechanics are established. By solving these equations, we can get the stress and strain in different plastic states.
The basic experiments of plastic mechanics are mainly divided into two categories: uniaxial tensile experiments and hydrostatic pressure experiments. The stress-strain curve, elastic limit and yield limit can be obtained by uniaxial tensile test. In the plastic state, the stress-strain relationship is nonlinear and there is no single-value correspondence. According to the hydrostatic pressure experiment, hydrostatic pressure can only cause elastic deformation of metal materials, and has little effect on the yield limit of materials (different geotechnical materials).
In order to simplify the calculation, according to the experimental results, the basic assumptions used in plastic mechanics are as follows: ① materials are isotropic and continuous. ② The average normal stress does not affect the yield of the material, but is only related to the volume strain of the material, and the volume strain is elastic, that is, the hydrostatic pressure state does not affect the plastic deformation but only produces the elastic volume change. This hypothesis is mainly based on the famous Brid-gman test. ③ The elasticity of materials is not affected by plastic deformation. These assumptions usually apply to metallic materials; For geotechnical materials, the influence of average normal stress on yield should be considered.
There are usually five simplified models for the stress-strain curve of plastic mechanics: ① ideal elastic-plastic model, which is used for low carbon steel or materials with unclear strengthening properties. (2) The linear strengthening elastic-plastic model is used to strengthen materials with obvious properties. ③ The ideal rigid-plastic model is used for materials with elastic strain much less than plastic strain and no obvious strengthening characteristics. ④ The linear hardening rigid-plastic model is used for materials with obvious hardening characteristics whose elastic strain is much less than plastic strain. ⑤ Power hardening model, in order to simplify the analytical formula in calculation, the analytical formula of stress-strain relationship can be written as σ = σy (ε/εy) n, where σy is the yield stress, εy is the strain corresponding to σy, and n is the material constant.
Yield condition and constitutive relation In complex stress state, the criterion for judging the yield state of an object is called yield condition. The yield condition is the condition that each stress component combination should meet. For metal materials, the most commonly used yield conditions are the maximum shear stress yield condition (also called tresca yield condition) and the elastic deformation specific energy yield condition (also called mises yield condition). Tresca yield condition, Drucker-praeger yield condition and Mohr-Coulomb yield condition are commonly used in geotechnical materials. For strengthened or softened materials, the yield condition will change with the increase of plastic deformation, and the changed yield condition is called subsequent yield condition. When the principal stress order is known, it is convenient to use tresca yield condition. If the order of principal stress is unknown, it is convenient to use Mises yield condition. For materials with good toughness, mises yield conditions are in good agreement with experimental data.
Because plastic deformation is related to deformation history, it is convenient to give the constitutive relation reflecting plastic stress-strain relationship in the form of strain increment. The theory of expressing plastic constitutive relation in the form of strain increment is called plastic increment theory. The constitutive relation of incremental theory is reasonable in theory, but it is troublesome in application, because the final result can only be obtained by integrating the whole deformation path. Therefore, the theory of total plasticity is developed, that is, the theory of expressing plastic constitutive relationship with total stress and total strain. Under the condition of proportional deformation, the constitutive relation of total quantity theory can be obtained by the constitutive relation of integral increment theory. When there are few out-of-proportion deformation conditions, the calculation result of total amount theory is close to the actual risk result. When solving the boundary value problem of plastic mechanics, the equilibrium equation, geometric equation (that is, the relationship between strain and displacement) and the boundary conditions of force and displacement are the same as those used in elasticity, but the physical relationship is no longer Hooke's law in elasticity, but the constitutive relationship of plastic increment or total amount is adopted.
The commonly used solution methods of plastic mechanics: ① statically determinate method. A solution to simple elastic-plastic problems. Because the number of unknowns is the same as the number of known equations, the unknowns in the problem can be found by applying the equilibrium equation and yield condition. ② Slip line method. It is suitable for solving plastic plane strain problems, and the stress components and corresponding displacement components of each point in the deformed body can be obtained. ③ Boundary method. A practical method, also known as the upper and lower bound method. The upper bound method uses the external work equal to the internal dissipation energy and the geometric conditions of the structure to find the plastic limit load, which is greater than the plastic limit load of the complete solution. The lower bound rule uses the equilibrium condition, yield condition and force boundary condition to find the plastic limit load, and its value is less than that of the complete solution. ④ Principal stress method. The contribution of shear stress is not considered in the yield condition, and the distribution of principal stress along a certain axis is assumed to be uniform. The distribution law of each stress component can be obtained by this method. ⑤ Parameter equation method. When Mises yield condition is used, the parametric equation satisfying the yield condition can be substituted into the equilibrium equation to solve it. ⑥ Weighted residual method. Mathematical methods for solving approximate solutions of differential equations. The main points are as follows: firstly, suppose a trial function as an approximate solution and substitute it into the control equation and boundary conditions of the required solution; Generally, this function cannot fully meet these conditions, so there is an error, which is the residual; By selecting a certain weight function and multiplying it by residue, and listing the algebraic equation to eliminate residue in the solution domain, the solution of differential equation can be transformed into a numerical calculation problem for solving algebraic equation, and an approximate solution can be obtained. ⑦ Finite element method. Elastic-plastic finite element method and rigid-plastic finite element method are commonly used, and the distribution law of stress and strain in deformed body can be obtained.
The main applications of plastic mechanics are: ① plastic limit analysis and stability analysis of structures, and complete solutions have been obtained for beams, trusses, rigid frames, arches, bent frames, circular plates, rectangular rods, cylindrical shells, spherical shells, conical shells and composite shells. (2) Plastic limit analysis and stability analysis of members, and the plastic limit loads of various tensile, flexural and torsional shafts and members with notches, grooves and holes are obtained. ③ Sheet metal forming, including drawing, flanging, expanding and necking. (4) Metal block forming, including upsetting, drawing, extrusion, forging and other processes. ⑤ Metal rolling, in which the metal material passes between two rollers rotating in opposite directions, and plastic deformation occurs. ⑥ Plastic dynamic response and plastic wave have important applications in protection engineering, earthquake engineering, armor-piercing and penetration, high-speed forming, hypervelocity impact and explosion engineering. ⑦ Self-tightening technology can improve the elastic strength of thick-walled cylinder and prolong the fatigue life by generating beneficial residual stress in the structure. ⑧ In geotechnical mechanics, it is used to study the bearing capacity of foundation, slope stability, the function of retaining wall and the bearing capacity of coal pillar. Pet-name ruby is used to study the method of estimating and eliminating residual stress.
Because the traditional plastic mechanics is only applicable to the plastic range of metals, especially hard metals, some basic concepts need to be revised when applied to rocks, soil, concrete and other materials, not only the development of generalized plastic mechanics. Generalized plastic mechanics abandons these assumptions, adopts component theory, and directly deduces plastic formula from solid mechanics principle, which is applicable to both geotechnical materials and metals.
The above mainly introduces the phenomenological study of plastic deformation based on experiments from a macro perspective. On the micro scale, micromechanics has been established, and its main research purpose is to establish the quantitative relationship between microstructure and mechanical properties from the material physics theory (dislocation, crystal paradigm, interface, etc.). ). Mesomechanics transforms the theoretical framework of classical continuum mechanics, introduces physical or geometric quantities to characterize the damage of material microstructure, and determines its evolution equation. At the same time, the homogenization method from meso to macro is developed, and the quantitative relationship between meso-structure, internal defects and macro-mechanical properties is established. Thereby forming a new theoretical framework on the meso-scale. The part of micromechanics related to plastic deformation is called plastic micromechanics. Compared with small deformation analysis of traditional plastic mechanics, Analysis of Large Plastic Deformation was written by Li Guochen and Mienne.
A brief history of plastic mechanics development
Plastic mechanics, as an important branch of solid mechanics, can be traced back to the 1970s at the earliest, but it was not fully developed and matured until the 1940s and early 1950s. Especially the ideal plasticity theory, has reached a mature stage and started to be applied to engineering practice. The phenomenon of plastic deformation was discovered earlier, but the mechanical study of it began with Coulomb 1773 earth pressure theory, and the yield condition of soil was put forward.
H.tresca put forward the maximum shear stress yield condition of metal materials in 1864. Subsequently, Saint Venant put forward the ideal rigid-plastic stress-strain relationship in the plane in 1870. He assumed that the direction of maximum shear stress and the direction of maximum shear strain rate were the same, and solved the problems of local plastic deformation in the cylinder and torsion and bending of internal pressure in the thick-walled cylinder. In 187 1, Levy extended the plastic stress-strain relationship to the three-dimensional case. 1900, the maximum shear stress yield condition was preliminarily determined by the combined test of tensile and internal pressure of the thin tube.
In the following 20 years, many similar experiments were carried out, and various yield conditions were put forward, among which the most significant one was the yield condition proposed by Mises in 19 13 (hereinafter referred to as Mises condition) based on the requirement of mathematical simplification. Mises also independently put forward the plastic stress-strain relationship consistent with Levi (hereinafter referred to as Levi-Mises constitutive relationship). Taylor experiment of 19 13 and Lode experiment of 1926 explore the stress-strain relationship, which has proved that the Levi-mises constitutive relation is a first-order approximation of the real situation.
In order to better fit the experimental results, under the inspiration of Pelant, Royce proposed a three-dimensional plastic stress-strain relationship including elastic strain in 1930. At this point, the theory of plastic increment is initially established. But at that time, there were still many difficulties in solving specific problems with incremental theory. As early as 1924, Hench put forward the plastic aggregate theory, which was used by Dai Na and other Soviet scholars, especially Ilyushin, to solve a large number of practical problems because of its convenient application.
Although the general theory of plasticity is not suitable for the complex stress change process in theory, the calculated results are very close to the experimental results of plate instability. Therefore, the debate between the plastic increment theory and the plastic aggregate theory started around 1950, which prompted the two theories to discuss on a more fundamental theoretical basis. In addition, in the study of strengthening law, in addition to the isotropic strengthening model, praeger also proposed the following strengthening model. The development of computer has opened up a broad prospect for the research and application of plastic mechanics, especially for the application of finite element method. In 1960, Argyris proposed that the initial load method can be used as the basis for solving elastic-plastic problems by finite element method. Since then, plastic mechanics with ideal plasticity has reached the finalization stage, while plastic mechanics with work hardening is still a developing research topic.
After the 1960s, with the development of finite element method, providing appropriate constitutive relation has become the key to solve the problem. Therefore, the research on plastic constitutive relation was very active in 1970s, mainly from the combination of macro and micro, from thermodynamics of irreversible process, and from rational mechanics.
In the aspect of experimental analysis, photoplasticity method, moire method and speckle interferometry are also used to measure large deformation. In addition, due to the plastic mechanical problems of rock materials, unstable materials with plastic volume strain, anisotropy, heterogeneity, elastic-plastic coupling and strain weakening are being studied.